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Abstract

We consider the obstacle problem for the Gauss curvature flow with an exponent \(\alpha \). Under the assumption that both the obstacle and the initial hypersurface are strictly convex closed hypersurfaces and that the obstacle is enclosed by the initial hypersurface, uniform estimates are obtained for several curvatures via a penalty method. We also prove that when the hypersurface is two dimensional with \(0<\alpha \le 1\), the solution of the Gauss curvature flow with an obstacle exists for all time with bounded principal curvatures \(\{\lambda _i\}\), where the upper bound is uniform, and the lower bound depends on the distance from the free boundary. Moreover, we show that there exists a finite time \(T_*\) after which the solution becomes stationary in the same shape as the obstacle.

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References

  1. Andrews, B.: Gauss curvature flow: the fate of the rolling stones. Invent. Math. 138(1), 151–161 (1999)

    Article  MathSciNet  Google Scholar 

  2. Andrews, B., Guan, P., Ni, L.: Flow by powers of the Gauss curvature. Adv. Math. 299, 174–201 (2016)

    Article  MathSciNet  Google Scholar 

  3. Brendle, S., Choi, K., Daskalopoulos, P.: Asymptotic behavior of flows by powers of the Gaussian curvature. Acta Math. 219(1), 1–16 (2017)

    Article  MathSciNet  Google Scholar 

  4. Choi, K., Daskalopoulos, P., Kim, L., Lee, K.A.: The evolution of complete non-compact graphs by powers of Gauss curvature. J. reine angew, Math (2016)

    MATH  Google Scholar 

  5. Chow, B.: Deforming convex hypersurfaces by the \(n\) th root of the Gaussian curvature. J. Differential Geom. 22(1), 117–138 (1985)

    Article  MathSciNet  Google Scholar 

  6. Crandall, M.G., Ishii, H., Lions, P.L.: User’s guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. 27(1), 1–67 (1992)

    Article  MathSciNet  Google Scholar 

  7. Firey, W.J.: Shapes of worn stones. Mathematika 21(1), 1–11 (1974)

    Article  MathSciNet  Google Scholar 

  8. Friedman, A.: Variational Principles and Free-Boundary Problems. John Wiley and Sons, New York (1982)

    MATH  Google Scholar 

  9. Hamilton, R.: Worn stones with flat sides. Discourses Math. Appl. 3, 69–78 (1994)

    MathSciNet  MATH  Google Scholar 

  10. Hamilton, R.S.: Three-manifolds with positive Ricci curvature. J. Differ. Geom. 17(2), 255–306 (1982)

    Article  MathSciNet  Google Scholar 

  11. Juutinen, P.: On the definition of viscosity solutions for parabolic equations. Proc. Am. Math. Soc. 129(10), 2907–2911 (2001)

    Article  MathSciNet  Google Scholar 

  12. Kim, L., Lee, K.A.: \(\alpha \)-Gauss curvature flows. arXiv preprint arXiv:1306.1100 (2013)

  13. Kim, L., Lee, K.A., Rhee, E.: \(\alpha \)-Gauss curvature flows with flat sides. J. Differ. Equ. 254(3), 1172–1192 (2013)

    Article  MathSciNet  Google Scholar 

  14. Tso, K.: Deforming a hypersurface by its Gauss–Kronecker curvature. Comm. Pure Appl. Math. 38(6), 867–882 (1985)

    Article  MathSciNet  Google Scholar 

  15. Zhu, X.P.: Lectures on mean curvature flows, AMS/IP Studies in Advanced Mathematics, vol. 32. American Mathematical Society, Providence (2002)

    Book  Google Scholar 

Download references

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Correspondence to Taehun Lee.

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Communicated by P. Topping.

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The authors would like to thank the anonymous referee for carefully reading the manuscript and for valuable comments that contributed to improve the presentation. Ki-Ahm Lee has been supported by Samsung Science and Technology Foundation under Project Number SSTF-BA1701-03, and holds a joint appointment with the Research Institute of Mathematics of Seoul National University. Taehun Lee was supported by National Research Foundation of Korea Grant (NRF-2014H1A2A1018664) funded by the Korea government and has been supported in part by a KIAS Individual Grant (MG079501) at Korea Institute for Advanced Study.

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Lee, KA., Lee, T. Gauss curvature flow with an obstacle. Calc. Var. 60, 166 (2021). https://doi.org/10.1007/s00526-021-02029-y

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  • DOI: https://doi.org/10.1007/s00526-021-02029-y

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